Tran Cao Son Department of Computer Science
New Mexico State University [email protected]
1 Invited Abstract
In this talk, I will detail the development of CpA(H), a competitive conformant planner, that won the Conformant Planning Category in the International Planning Competition 2006. Lessons learned and the influence of logic programming in our development of the planner will also be discussed.
Conformant Planning with Disjunctive Initial States:
Design and Development of an Efficient Planner
D-V. Tran, H-K. Nguyen, E. Pontelli, T.C. Son Department of Computer Science
New Mexico State University
vtran | knguyen | epontell | [email protected]
Abstract. The paper illustrates the design and development of a competitive con- formant planner. The planner builds on the theoretical foundations of approximation- based planning to enable a compact representation of the possible states. The novelty of the proposed approach is the realization that the description of the (incomplete) initial state is often based onconstraints(e.g., expressed through PDDL’sorandoneofclauses); the paper illustrates how such constraints can be reasoned upon to reduce the size of the search space. The reasoning process is implemented in the form of transformations of the problem specification within the planner. This, along with approximations and the use of combined heuris- tics, leads to enhanced efficiency and scalability, outperforming state-of-the-art conformant planners on several benchmark suites.
1 Introduction and Motivation
Conformant planning is the problem of finding a sequence of actions that achieves the goal from every possible initial state of the world [14]. One of the main difficulties encountered in the process of determining a conformant plan is the high degree of un- certainty, due to the potentially large number of possible initial states of the problems.
The Planning Domain Definition Language (PDDL)introduces two constructs to express incomplete knowledge about the initial state of the world:mutual-exclusion statements (expressed usingoneof-clauses) anddisjunctivestatements (expressed us- ingor-clauses). Frequently,oneof-clauses are used to specify the possible initial states and or-clauses are used to eliminate infeasible states. Because of this, the number of possible initial states depends mainly on the number and the size of the oneof- clauses—and these are often exponential in the number of constants present in the prob- lem instances. For example, three out of six domains in the 2006 planning competition have this property (Table 1).
Instance # Cons/States Instance # Cons/States comm-15 35/216 coins-20 17/9×86 comm-20 85/221 coins-25 39/1020 comm-25 140226 coins-30 45/1025 sortnet-10 11/211 sortnet-15 216/16 Table 1.Number of Constants/Possible Initial States
Effective methodologies and data structures are required to deal with the large number of possible initial states. Some conformant planners, such as POND [6] and KACMBP [7], employ a BDD representation of belief states, while others, such as CFF [4], adopt a CNF representation. These types of encodings avoid dealing directly with the exponential number of states, but they require extra work in determining the truth value of certain fluents after the execution of a sequence of actions in the initial belief state. For instance, CFF needs to make a call to a SAT-solver with the initial state and the sequence of actions; other planners need to recompute the BDD representation, which could also be an expensive operation. Observe that the problem of determining the truth value of a proposition after the execution of a single action in a belief state is co-NP complete [1].
An alternative approach to deal with the large number of possible initial states is used by the plannerscf2cs(ff)and CPA [12, 17], and further investigated in their successorst0and CPA+ [13, 16]. This approach relies on anapproximation seman- ticsin reasoning with incomplete information [15]. The plannerscf2cs(ff)andt0 reduce the number of possible initial states to one by introducing additional proposi- tions, transforming the original problem to a classical planning problem, and using FF, a classical planner [8], to find solutions. On the other hand, CPA and CPA+ reduce this number by dividing them into groups and using the intersection of each group as its representative during planning.
CPA+ andt0implement the idea of approximations differently. While CPA+ could be seen as a standard heuristic search forward planner,t0follows a translational ap- proach. The performance of CPA+ depends on its heuristic function and its ability to approximate the initial belief state to a manageable set of partial states. On the other hand, the performance oft0largely depends on the performance of FF.t0was the winner of the 2006 planning competition.
In this paper, we describe the design and implementation of a competitive confor- mant planner. The proposed planner1 expands the idea of approximation-based con- formant planning, introducing novel techniques to significantly enhance efficiency and scalability. We explore the problem of engineering an approximation-based planner that can avail of modern data structures and heuristic functions. A cornerstone of our ap- proach is viewing the description of the initial state not just as a passive characteriza- tion of a collection of states, but as a collection of constraints; by reasoning on such constraints, we discover ways to transforms the problem specification, enabling drastic reductions in the size of the search space. The resulting planner outperforms the state- of-the-art in conformant planning on large pool of benchmarks, including the problems from the latest planning competition.
2 Problem Representation
Following the notation in [12], we describe a problem specification as a tupleP = hF, O, I, Gi, whereF is a set of propositions,Ois a set of actions,IandOdescribe the initial state of the world and the goal respectively. Aliteralis either a proposition p ∈ F or its negation¬p.`¯denotes the complement of a literal`and is defined by
1Named CPA(H) to recognize its roots in the CPA+ system.
`¯=¬`where¬¬p=pforp∈F. For a set of literalsL,L={`¯|`∈L}. A conjunction of literals is often represented by a set.
A set of literalsX isconsistentif there exists nop ∈ F such that{p,¬p} ⊆ X.
Astatesis a consistent andcompleteset of literals, i.e.,sis consistent, and for each p ∈ F, eitherp ∈ sor ¬p ∈ s. Abelief stateis a set of states. Apartial stateis a consistent set of literals. Acs-stateis a set of partial states. A set of literalsX satisfies a literal`(resp. a set of literalsY) iff`∈X (resp.Y ⊆X).
Each actionainO is associated with a preconditionφ(denoted bypre(a)) and a set of conditional effects of the formψ→`(also denoted bya:ψ→`), whereφand ψare sets of literals and`is a literal.
The initial state of the world is described byI = Id∪Io∪IrwhereId is a set of literals,Io is a set ofoneof-clauses—of the formoneof(φ1, . . . , φn)—andIr is a set ofor-clauses of the formor(φ1, . . . , φn), where each φi is a set of literals. A oneof-clause indicates that the φi’s are mutually exclusive, while anor-clause is a disjunctive normal form (DNF) representation of a formula. A set of literalsXsatisfies oneof(φ1, . . . , φn) if there exists some 1≤i≤ns.t. φi ⊆ X and for every j 6= i, 1≤j≤n,φj∩X6=∅.X satisfiesor(φ1, . . . , φn)if there exists some1≤i≤ns.t.φi ⊆ X. ext(I) denotes the set of all states satisfyingId, everyoneof-clause inIo, and everyor-clause in Ir. E.g., if F={g, f} andI={g,oneof(f,¬f)} then ext(I) = {{g, f},{g,¬f}}.
Gcan contain literals oror-clauses. Given aoneof-clause or anor-clauseo, we writeL∈oto denote thatLis an element ofoandlit(o) =S
L∈o(L∪L).¯
3 A Competitive Conformant Planner: Design
The planner, called CPA(H), is composed of two modules. The first module (Prepro- cessor) is a static analyzer that performs a number of transformations of the problem specification. Along with a grounder (which also applies standard simplifications, such as forward reachability), the preprocessor applies some novel transformations (oneof- clause combination and goal splitting) aimed at drastically reducing the size of the search space. The second module (Planning engine) is a heuristic search engine imple- menting forward planning.
3.1 Design of the Planning Engine
Theoretical Foundations Given a state s and an action a, a is executable in s if pre(a) ⊆ s. The set of effects of ains, denoted by ea(s), is defined by: ea(s) = {l |ψ→lis an effect ofa, ψ⊆s}.The execution ofain a statesresults in a succes- sor statesucc(a, s)which is defined by:succ(a, s) =s∪ea(s)\ea(s)ifais executable ins; andsucc(a, s) =failed, otherwise.
succis extended to definesucc∗, which computes the result of the execution of an action in a belief state, as follows.
succ∗(a, S) =
{succ(a, s)|s∈S}
ifais executable in everys∈S
failed otherwise (1)
Finally, we can define the functionsuccd to compute the final belief state resulting from the execution of a plan:
• succ([ad 1, . . . , an], S) =Sifn= 0, and
• succ([ad 1, . . . , an], S)=succ∗(an,succ([ad 1, . . . , an−1], S))ifn >0.
Several heuristic search-based conformant planners (e.g. CFF, POND), employsuccd in plan computation, usingS0=ext(I)as the initial belief state. An action sequenceα is asolutionofPiffsucc(α, Sd 0)6=failedandGis satisfied in every state belonging tosucc(α, Sd 0).
The notion of approximation used in CPA+,cf2cs(ff), andt0has been origi- nally proposed in [15]. The original intuition behind approximation is to approximate sets of possible states by a single partial state—thus, reducing the complexity of reason- ing w.r.t. using all possible states. It is characterized by a function (succA) that maps an action and a partial state to a partial state. Thepossible effectsofain a partial state δare given by
pca(δ) ={l|ψ→lis an effect ofa, ψ∩δ=∅}. (2) The successor partial state from the execution ofainδis defined bysuccA(a, δ) = (δ∪ea(δ))\pca(δ)ifais executable inδ; andsuccA(a, δ) =failed, otherwise.
Similarly tosucc∗andsucc,d succAcan be extended to definesucc∗A(mapping cs- states to cs-states) andsucc\A for computing the result of the execution of an action sequence starting from a cs-state. The notion of a solution ofPw.r.t. the approximation is extended accordingly. Our planner usessucc\∗Ain its search for plans.
Observe that, in general, reasoning using approximations is incomplete. Complete- ness can be gained by identifying appropriate partitions{∆1, . . . , ∆k}ofext(I)such thatSk
i=1∆i=ext(I),∆i∩∆j=∅for eachi6=j, and we have that for each formula ϕand sequence of actionsα,succ\A(α,{δ1, . . . , δk})entailsϕiffsucc(α, ext(I))d en- tailsϕ, whereδiis the intersection of the states in∆i. Research has been conducted to provide sufficient syntactical conditions to identify valid partitions—based on the iden- tification of fluents that should be explicitly distinguished in different partitions (see, e.g., [16]).
Heuristic Search Thesucc∗A function is used in the context of a planning algorithm which implements forward planning using a traditional best-first heuristic search.
The proposed planner enables the user to choose among different heuristics; as dis- cussed in the experimental section, we observed that sustained better performance can be achieved by using combinations of heuristics, improving the ability of discriminat- ing between states in the priority queue (as employed in other systems as well [6]). The basic heuristics employed are:
• The cardinality heuristic: we prefer belief states that have a smaller cardinality.
In other words,hcard(Σ) =|Σ|whereΣ is a belief state. Note that we use this heuristic in a forward fashion, and this is different from its use in [2, 5]. The intu- ition behind this is that planning with complete information is “easier” than plan- ning with incomplete information and a lower cardinality implies a lower degree of uncertainty.
• The relaxed graphplan heuristic:for a belief stateΣ, we definehrgp(Σ) =P
δ∈Σd(δ), where d(δ)is the well-known sum heuristic value given that the initial state is δ∪ {¬p|p∈F, p6∈δ, ¬p6∈δ}[11].
• The number of satisfied subgoals:denoted byhgc(Σ).
We investigate the following combination of these heuristics:hcss(Σ) = (hcard(Σ), hgc(Σ), hrgp(Σ));
heuristic measures are compared according to their lexicographic order.
3.2 Design of the Preprocessor
Standard Transformations Key to our analysis is the notion ofdependencebetween actions and propositions—similar to the notion of dependence between actions and literals explored in [16]. We denote withP a planning problem.
Definition 1. An actionadependson a literal`if 1. `∈pre(a), or
2. there exists an effecta:φ→hinPand`∈φ, or
3. there exits an actionbthat depends on`andadepends on some of the effects ofb, i.e.,bdepends on`and there existsb:φ→hsuch thatadepends onh.
Bypreact(`)we denote the set of actions depending on`.
Intuitively, the fact thatadepends on`indicates that the truth value of`could influence the result of the execution ofa. For a set of literalsL,preact(L) =S
`∈Lpreact(`).
Definition 2. Two literals`and`0 aredistinguishableif`6=`0and there is no action that depends on both`and`0, i.e.,preact(`)∩preact(`0) =∅.
Obviously, the distinguishable relation is symmetric and irreflexive. Two set of literals L1andL2are distinguishable ifpreact(L1)∩preact(L2) =∅.
The dependence between a literal and an action, often used inreachability analysis, is defined next.
Definition 3. A literal`dependson an actionaif (1)a:ψ→`is inP; or(2)there exists an actionbsuch thatb:ψ→`is inP and there exists some`0inψor inpre(b) s.t.`0depends ona. We denote withdeps(a)the set of literals that depend ona.
Intuitively,`depends onaimplies that`may be achieved by executinga.postact(`) denotes the set of actions which`depends on, i.e.,postact(`)={a|`∈deps(a)}.
Definition 4. Two literals`and`0areindependentif`6=`0and there exists no action that both`and`0depend on, i.e.,postact(`)∩postact(`0) =∅.
The preprocessor starts its operations with a number of basic normalization steps, aimed at reducing the number of propositions and the number of actions present in the problem specification. In particular, it implements a traditionalforward reachability simplification (to detect propositions whose value cannot be changed and actions that cannot be triggered w.r.t. the initial state) and a symmetricalgoal relevance(removing actions that cannot contribute to the goal).
Combination ofoneof-clauses The idea of this technique is based on thenon-interaction between actions and propositions in different sub-problems of a conformant planning problem. We illustrate this idea in the next example.
Example 1. LetP =h{f, g, h, p, i, j}, O, I, GiwhereI={oneof(f, g),oneof(h, p),¬i,¬j}, G=i∧j, andO =©
a:f →i c:h→j b:g→i d:p→jª
. It is easy to see that the sequenceα = [a, b, c, d]is a solution ofP. Furthermore, the search should start from the initial belief state consisting of the four states:
{f,¬g, h,¬p,¬i,¬j} {¬f, g, h,¬p,¬i,¬j}
{f,¬g,¬h, p,¬i,¬j} {¬f, g,¬h, p,¬i,¬j}
LetP0be the problem obtained fromPby replacingIwithI0, whereI0={oneof(f∧ h, g∧p),¬i,¬j}.
We can see that α is also a solution of P0. Furthermore, each solution of P0 is also a solution ofP. This transformation in interesting since the initial belief state now consists only of two states:{f,¬g, h,¬p,¬i,¬j}and{¬f, g,¬h, p,¬i,¬j}. I.e., the number of states in the initial belief state that a conformant planner has to consider in P0is2, while it is4inP. This transformation is possible because the set of actions that are “activated” byf andgis disjoint from the set of actions that are “activated” byh andp, i.e.,preact({f, g})∩preact({h, p}) =∅.
The above example shows that differentoneof-clause can be combined into a single oneof-clause, which effectively reduces the size of the initial state that a planner needs to consider in its search for a solution. Theoretically, if the size of the twooneof-clauses in consideration ismandn, then it is possible to achieve a reduction in the number of possible partial states from m×n tomax(m, n). Since in many problems the size of theoneof-clauses increases with the number of objects, being able to combine the oneof-clauses could provide a significant advantage for the planner.
Definition 5. Let P = hF, O, I, Gi be a planning problem. Two oneof-clauses o1 ando2 arecombinableif (i)lit(o1)∩lit(o2) = ∅; and (ii)lit(o1)is distinguishable fromlit(o2). wherelit(o)denote the union of the set of literals occurring inoand its complements
For example, theoneof-clauses in Ex. 1 are combinable. Leto1=oneof(L1, . . . , Ln)ando2= oneof(S1, . . . , Sm). Assume thatn ≥ m. Acombinationof o1 ando2, denoted by o1⊕o2(oro2⊕o1) is the clause
oneof(L1∧S1, . . . , Lm∧Sm, Lm+1∧S1, . . . , Ln∧S1)
Intuitively, a combination ofo1 ando2 is a oneof-clause whose elements are pairs obtained by composing one element ofo1with exactly one element ofo2.
Proposition 1. LetP = hF, O, I, Gibe a planning problem, whereGis a conjunc- tion of literals and o1 and o2 are two combinable oneof-clauses in P. Let P0 = hF, O, I0, Gi, whereI0 is obtained fromI by replacingo1 ando2 byo1⊕o2. Every solution ofP0is also a solution ofPand vice versa.
Observe that the above proposition may not hold ifP contains disjunctive goals, as shown next.
Example 2. LetP =h{q, g, h, p, i, j}, O, I, Giwhere
I={oneof(h, g),oneof(p, q),¬i,¬j}andG=or(i, j) andOconsists ofa:p,¬q→i,c:p, q→i,b:g,¬h→j, andd:g,¬h→j.
It is easy to check thatoneof(h, g)andoneof(q, p)are combinable. LetP0be the problem obtained fromPby replacingIwithI0={oneof(g∧q, h∧p),¬i,¬j}. Then, [a, b]is a solution ofP0but not a solution ofP.
Thecombinablenotion can be generalized as follows.
Definition 6. A set ofoneof-clauses{o1, . . . , ok}is combinable ifoiandojare com- binable for each1≤i6=j≤k.
Let⊕(o1, . . . , ok)be the shorthand for(((o1⊕o2)⊕. . .)⊕ok). Proposition 1 can be generalized as follows.
Proposition 2. LetP = hF, O, I, Gibe a planning problem, whereGis a conjunc- tion of literals. Let{o1, . . . , ok}be a combinable set ofoneof-clauses inP. LetP0= hF, O, I0, Gi, whereI0is obtained fromIby replacing{o1, . . . , ok}with⊕(o1, . . . , ok).
We have that each solution ofP0is a solution ofP and vice versa.
We implemented a greedy algorithm, whose running time is polynomial in the size of P, for detecting sets of combinableoneof-clauses and replacing them with their corresponding combination. This is possible since testing if`and`0are distinguishable can be done in polynomial time in the size ofP, and the number of pairs that need this test is quadratic in the number of propositions.
Goal Splitting Reducing the size of the initial state only helps the planner to start the search. It does not necessarily imply that the planner can find a solution. In this section, we present another technique, calledgoal-splitting, which can be used in conjunction with the combination ofoneofto deal with large planning problems. This technique can be seen as a variation of the goal ordering technique in [9] and it relies on the notion of dependence proposed in Def. 4. The key idea is that if a problemPcontains a subgoal whose truth value cannot be negated by the actions used to reach the other goals, then the problem can be decomposed into smaller problems with different goals, whose solutions can be combined to create a solution of the original problem. This is illustrated in the following example.
Example 3. Consider the problemPof Example 1. It is easy to see that the two goalsi andjare independent andPcan be decomposed into two sub-problemsP1=hF, O1, I, ii andP2 = hF, O2, I2, jiwhereO1 = {a : f →i, b : g →i}andO2 ={c : h→ j, d:p→j}with the following property: ifαis a solution ofP1andβis a solution ofP2whereI2=succ\A(α, I1), thenα;βis a solution ofP.2
Let us start with a definition capturing the condition that allows the splitting of goals.
2α;βdenotes the concatenation of two sequences of actions.
Definition 7. LetP =hF, O, I, Gibe a planning problem and let`∈G. We say that
`isG-separable if, for each`0∈G\ {`}we have that`¯and`0are independent.
Proposition 3. LetP =hF, O, I, Gibe a planning problem and let`beG-separable.
LetP`=hF, postact(`), I, `iandαbe a solution ofP`. LetPG\{`}=hF, postact(G\ {`}), I0, G\ {`}i, whereI0 =succ\A(α, I), andβ be a solution ofPG\{`}. Then,α;β is a solution ofP.
The proof is trivial, sincepostact(G\ {`})does not contain any action that can make
`¯true.
On the other hand, it is easy to see that not every plan ofP can be split into two partsαandβsuch thatαis a solution ofP`andβis a solution ofPG\{`}. We can prove, however, that for each planγofP, there is a planαforP`and a planβforPG\{`}such thatγis a permutation ofα;β. This provides a weak form of completeness.
We note that the splitting proposed in Prop. 3 can be improved by also splitting the propositions and initial states into different theories. We have implemented a general- ized version of Prop. 3 to split a problem into a sequence of problems. This implemen- tation runs in polynomial time in the size ofP.
4 Implementation Considerations
The preprocessor has been implemented as a Prolog program. The program maps the input PDDL theory to a collection of Prolog clauses. This mapping nicely avoids the need of explicitly grounding the problem specification a priori. The transformations are implemented as fixpoint computations on the Prolog clauses representing the problem specification.
The planning engine has been implemented as a C++ program, running on a Linux (Athlon 64, 4Ghz), gcc 4.2.1 version, with STL library. A partial state is implemented as a set (a basic data structure in STL) of literals.
The engine implements a best first search over the search space of cs-states. Each cs-state is a data structure consisting of a set of partial states, a plan to reach that cs- state, and the heuristic values:hcard,hgc, andhrpg. A modified version of the algorithm presented in [10] is implemented to computehrpg.
succ∗A is used to compute the next cs-state. A hash table (resp. priority queue) is used to store the visited (resp. unvisited) cs-states. A special module is developed to compute the initial cs-state, which consists of the set of initial partial states. Each initial partial stateδsatisfies the following conditions: a){p,¬p}∩δ6=∅for each proposition pappears inI; b)Id ⊆δ; c) for eachoneof(φ1, . . . , φn)∈Io, there exists anisuch thatφi⊆δand for allj6=i,φj∩δ6=∅; d) for eachor(φ1, . . . , φn)∈Ir, there exists anisuch thatφi ⊆δ; e)δis consistent. Choosing to implement the initial cs-state as a set (of the set of initial partial states) makes the computation of the successor cs-state (the result ofsucc∗A) easier. The main disadvantage of this choice is that the size of the initial cs-state can be exponential in the size of the number of object constants in the problem. This is the reason why reducing the size of the initial cs-state is critical to our planner.
5 Experimental Evaluation
The experimental evaluation has been performed using several benchmark suites—i.e., problems from the IPC-5 (or I5) and the IPC-6 (I6) planning competitions, challeng- ing (C) problems proposed in [13], and several other domains from previous planning competitions. The benchmark suite for each domain is listed in Table 5. Due to lack of space, we omit the detailed description of the actual benchmarks, that have been drawn from the existing literature. We also report only a subset of the complete experimental results due to limited space (full results will be made available through a linked tech- nical report). Time is in seconds, TO denotes time-out (30 min), AB denotes out of memory, and BM denotes benchmark suite.
Table 2 summarizes some results aimed at evaluating the impact of the transforma- tions; the three columns indicate execution times and the length of the first plan found;
we can observe that the improvement in performance is often significant; it occasionally comes at the price of a longer plan.
Problem NoTransf. oneofgoal-splitting coins-05 0.02/13 0.024/19 0.03/14 coins-10 1.33/35 0.66/129 1.52/43
coins-15 /AB 13.54/391 /AB
coins-20 /AB 33.39/621 /AB
comm-05 0.90/48 0.31/60 0.20/35 comm-10 61.14/87 3.98/190 22.40/65
comm-15 /AB 15.82/327 /AB
uts-05 34.43/115 34.43/115 1.06/43 uts-10 36.14/130 36.14/130 21.21/87 uts-20 53.88/286 53.88/286 35.91/138 uts-30 152.07/177 152.07/177 18.06/74 dispose-4-2 1.30/72 0.395/59 0.78/76 dispose-4-3 43.00/93 0.36/78 19.14/111 dispose-8-2 412.70/272 10./234 368.03/284 dispose-8-3 /AB 487.28/1187 /AB push-4-2 1.11/58 1.00/133 1.84/96 push-4-3 34.66/265 2.04/251 46.68/141 push-8-2 282.55/444 128.21/979 /AB
push-8-3 /AB 454.80/1811 /AB
Table 2.Impact ofoneof-combination and goal-splitting
Table 3 reports the execution times of the planner using different heuristics. The column tS indicates the time for preprocessing. Although the results are mixed for small instances,hcss(Σ)comes ahead for large instances.
Table 4 compares the execution times and plan lengths for the proposed planner and other three state-of-the-art systems for conformant planning (t0, CFF, and POND, all run with default parameters according to their documentation). Table 5 reports the number of instances each planner can solve.
6 Discussion
We now discuss the question of whether the proposed techniques can be applied to other planning systems.
Observe that a combination of severaloneof-clauses is aoneof-clause, whose ele- ments are conjunctions of literals, which can be represented by a set ofoneof-clauses
Problem tS CPA(H) CPA(H) CPA(H) CPA(H) hcard hgc hrgp hcss(Σ)
bwl-02 0.13 0.262/26 /TO 0.064/33 0.217/41
bwl-03 0.18 7.668/198 /TO 2.219/145 73.188/312 coins-15 0.49 7.449/423 15.874/551 0.387/191 5.920/329 coins-20 0.66 25.51/756 24.413/722 0.902/195 20.239/481 comm-15 3.64 0.496/96 0.148/95 0.094/95 0.124/95 comm-20 141 2.739/240 0.993/239 0.994/239 0.976/239 comm-25 1081 18.74/389 3.355/389 3.674/389 3.249/389 sortnet-05 0.11 0.054/13 16.35/13 0.036/12 0.023/12 sortnet-10 0.24 12.537/39 /TO 3.270/39 4.205/39
sortnet-15 0.52 /TO /TO /TO 313.044/65
uts-10 0.93 21.21/87 /TO 17.567/89 9.602/80 uts-20 0.89 35.91/138 /TO 12.596/150 36.314 /125 uts-30 0.86 18.06/74 /TO 346.467/103 31.609/94 d-4-3 .61 . 3.94/288 4.83/369 0.95/314 3.80/288 d-8-1 47.3 1.95/143 124.71/1229 24.12/725 1.86/137 d-8-2/C 53.6 386.68/1328 600.77/2298 135.8/1494 391.44/1328 d-10-1 285 7.65/213 /AB 148.7/1489 7.69/213 push-4-3 .65 40.45/1176 340.23/959 1.0/225 288.21/847 push-8-1/C 52.4 3.824/184 /AB 27.25/468 3.97/184
push-10/C 304 23.04/414 /AB /AB 23.48/414
1-d-4-3 .69 24.8/64 /AB 81.57/108 25.12/64
1-d-8-1 47.9 9.91/340 /AB /AB 9.22/340
1-d-10-1 288 36/568 /AB /AB 33.81/568
lng-8-1-1 50.4 2.45/94 284.73/5547 /AB 1.78/94 lng-8-2-1 57.8 0.97/94 31.79/563 68.45/287 0.97/94
lng-8-1-2 59.5 73.14/125 /AB /AB 72.65/125
Table 3.Comparison between heuristics
and a set of disjunctions eliminating the unwanted combinations. We tested the effec- tiveness of the oneof-simplification on POND and CFF. Table 6 shows the results of our experiment with the planner POND in thecommandcoinsdomains where theoneof-combination is applicable. As we can see, the performance of POND im- proves in these problems, and the improvement is more significant when the size of the problem is large. This technique helps POND to scale up but its impact is not as great as in CPA(H): POND can solve more problems in thecommdomain. The prob- lemscomm-16.0andcomm-16.1have more objects thancomm-16but less than comm-17. For CFF, we did not observe improvements using the modified problems.
We also experimented with using the preprocessor to perform goal splitting and produce modified PDDL files that can be processed by other planners. For example, CFF is unable to solve the problems from p21 to p30 of thecoinsdomain. The diffi- culty in this domain lies in the large number of elevators and coins. The goal-splitting technique divides the problem into a sequence of sub-problems, each dealing with one coin but still has all elevators, enabling CFF to solve these problems. We observed that CFF spent most of the time finding the solution for the first problem. This is reason- able, since the location of the elevators is initially unknown, and some locations will be known at the end of the first solution. The planning time of CFF for coins p21, p25 and p30 is 24.48 2.13 and 68.09 (secs) accordingly.
These initial experiments show that the proposed techniques could be useful for other planners.
7 Conclusions
In this paper, we presented the complete design and implementation of an efficient con- formant planner, called CPA(H). The planner builds on recently developed techniques for conformant planning using approximations; it introduces several novelties, includ- ing a preprocessing module to transform the problem specification, leading to signifi- cantly reduced search spaces, and the ability to explore the search space with different
Problem tS CPA(H) t0 CFF POND hcss(Σ)
block-03 0.13 0.22/41 /NA /AB 1.02/26
block-04 0.18 73.19/312 /NA /AB 1379/111 coins-10 0.18 0.14/81 0.09/26 1.02/38 5.26/28 coins-15 0.49 5.92/329 0.26/81 7.35/79 /TO coins-20 0.66 20.24/481 0.32/108 38.19/143 /TO comm-15 3.64 0.12/95 0.19/110 0.22/95 1662/110 comm-20 141 0.98/239 0.86/278 13.33/239 / TO comm-25 1081 3.25/389 3.99/453 109.49/389 / TO
sortnet-10 0.24 4.21/39 NA NA /TO
sortnet-15 0.52 313.04/65 NA NA /TO
adder-01 8.2 1.29/3 /NA /AB /AB
UTS-cycle-03 0.17 1.08/3 /NA /NA 1.99/3 UTS-cycle-04 0.29 23.88/6 /NA /NA 48.19/6 forest-02 2.4 23/84 0.37/12 0.03/17 1.33/15 forest-04 7.7 /TO 1.58/60 /TO 71.17/62 Rao-keys-02 0.22 0.04/32 /NA 0.08/33 0.29/21 Rao-keys-03 0.37 3.84/152 /NA 25.09/101 4.51/68 dispose-8-1 47.3 1.86/137 27.85/291 423.25/226 /AB dispose-8-2 53.6 391/1328 118.46/422 /AB /AB dispose-10-1 285 7.69/213 275.08/474 /AB /AB
Table 4.Comparison between systems (NA: not applicable) Domain/BM # of CPA(H) t0CFF POND
instanceshcss(Σ)
block/I6 4 3 0 1 3
adder/I5 4 1 0 0 0
coins/I5 30 20 20 20 10
comm/I5 25 25 25 25 16
sortnet/I5 15 15 0 0 6
uts/I5 30 30 30 30 24
UTS-cycle/I6 27 2 0 0 2
forest/I6 9 1 8 1 2
Rao-keys/I6 29 2 0 2 2
dispose/C 90 62 50 41 8
push/C 90 29 33 27 12
1-dispose/C 90 24 7 1 8
look-n-grab/C 81 63 20 21 9
Table 5.Number of problems solved in different domains
heuristic functions. The result is a conformant planner that has been shown to be highly competitive with the state-of-the-art in the field. In particular, CPA(H) outperforms all existing systems on the problems from the latest International Planning Competition.
The results presented in this paper confirm the strength of using approximations for conformant planning, the possibility of implementing approximation-based planning in an efficient and scalable system, and the scope for improvement that can be achieved via transformation of problem specifications.
Problem Orig/Modified Problem Orig/Modified comm-15 1662/4.89 coins-5 0.52/0.51 comm-16 TO/57.93 coins-10 5.54/1.63 comm-16.0 TO/124.05 coins-15 17.13/17.95 comm-16.1 TO/267.39 coins-10 143/126
Table 6.Impact ofoneof-Combination on POND (Orig/ Modified: Time for solving the origi- nal/modified problem)
The future developments of this project include exploring whether alternative meth- ods for the internal implementation of cs-states (e.g., OBDD) can further enhance per- formance. We also plan to expand the reasoning component of the preprocessor, to ob- tain additional simplifications of the problem specifications (e.g., detecting symmetries between fluents).
Acknowledgement
The authors are partially supported by the NSF grants IIS-0812267, CBET-0754525, CNS-0220590, and CREST-0420407.
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